| L(s) = 1 | − 4-s − 8·11-s + 16-s + 12·19-s − 8·29-s + 4·41-s + 8·44-s + 10·49-s + 8·59-s − 20·61-s − 64-s + 16·71-s − 12·76-s − 16·79-s − 28·89-s + 32·101-s − 24·109-s + 8·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2.41·11-s + 1/4·16-s + 2.75·19-s − 1.48·29-s + 0.624·41-s + 1.20·44-s + 10/7·49-s + 1.04·59-s − 2.56·61-s − 1/8·64-s + 1.89·71-s − 1.37·76-s − 1.80·79-s − 2.96·89-s + 3.18·101-s − 2.29·109-s + 0.742·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34222500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34222500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9093519332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9093519332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.284409498918478212573024667001, −7.76724301066102644703893092073, −7.50633066083979316107929889418, −7.49583382444940258429579924327, −7.07099076656885750425054321859, −6.53653017829304975585248196308, −5.87034818797974510518008619838, −5.65148674972398163299322406546, −5.39842198843960328438765979144, −5.21823898635676650945669841281, −4.69255021877772115978277856029, −4.39246975800362600192230329504, −3.67448519944382708422263868417, −3.54823700787849834052437616517, −2.83510649403262469451582481210, −2.76715665954371806895572288870, −2.23784057413232239015662363670, −1.50854464504898687054929078518, −1.01228458345248967313177461384, −0.28063209462906563364628391652,
0.28063209462906563364628391652, 1.01228458345248967313177461384, 1.50854464504898687054929078518, 2.23784057413232239015662363670, 2.76715665954371806895572288870, 2.83510649403262469451582481210, 3.54823700787849834052437616517, 3.67448519944382708422263868417, 4.39246975800362600192230329504, 4.69255021877772115978277856029, 5.21823898635676650945669841281, 5.39842198843960328438765979144, 5.65148674972398163299322406546, 5.87034818797974510518008619838, 6.53653017829304975585248196308, 7.07099076656885750425054321859, 7.49583382444940258429579924327, 7.50633066083979316107929889418, 7.76724301066102644703893092073, 8.284409498918478212573024667001
